US7486839B2 - Efficient method for MR image reconstruction using coil sensitivity encoding - Google Patents
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- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
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- G01R33/561—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution by reduction of the scanning time, i.e. fast acquiring systems, e.g. using echo-planar pulse sequences
- G01R33/5611—Parallel magnetic resonance imaging, e.g. sensitivity encoding [SENSE], simultaneous acquisition of spatial harmonics [SMASH], unaliasing by Fourier encoding of the overlaps using the temporal dimension [UNFOLD], k-t-broad-use linear acquisition speed-up technique [k-t-BLAST], k-t-SENSE
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- G01R33/28—Details of apparatus provided for in groups G01R33/44 - G01R33/64
- G01R33/285—Invasive instruments, e.g. catheters or biopsy needles, specially adapted for tracking, guiding or visualization by NMR
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- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
- G01R33/56—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution
- G01R33/561—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution by reduction of the scanning time, i.e. fast acquiring systems, e.g. using echo-planar pulse sequences
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- G—PHYSICS
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- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
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- G01R33/28—Details of apparatus provided for in groups G01R33/44 - G01R33/64
- G01R33/32—Excitation or detection systems, e.g. using radio frequency signals
- G01R33/34—Constructional details, e.g. resonators, specially adapted to MR
- G01R33/34084—Constructional details, e.g. resonators, specially adapted to MR implantable coils or coils being geometrically adaptable to the sample, e.g. flexible coils or coils comprising mutually movable parts
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/4818—MR characterised by data acquisition along a specific k-space trajectory or by the temporal order of k-space coverage, e.g. centric or segmented coverage of k-space
- G01R33/4824—MR characterised by data acquisition along a specific k-space trajectory or by the temporal order of k-space coverage, e.g. centric or segmented coverage of k-space using a non-Cartesian trajectory
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
- G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
- G01R33/48—NMR imaging systems
- G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
- G01R33/56—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution
- G01R33/5608—Data processing and visualization specially adapted for MR, e.g. for feature analysis and pattern recognition on the basis of measured MR data, segmentation of measured MR data, edge contour detection on the basis of measured MR data, for enhancing measured MR data in terms of signal-to-noise ratio by means of noise filtering or apodization, for enhancing measured MR data in terms of resolution by means for deblurring, windowing, zero filling, or generation of gray-scaled images, colour-coded images or images displaying vectors instead of pixels
Definitions
- MRI magnetic resonance imaging
- reconstruction method for sensitivity encoding with non-uniformly sampled k-space data.
- Magnetic resonance imaging is a diagnostic imaging modality that does not rely on ionizing radiation. Instead, it uses strong (ideally) static magnetic fields, radio-frequency (“RF”) pulses of energy and magnetic field gradient waveforms. More specifically, MR imaging is a non-invasive procedure that uses nuclear magnetization and radio waves for producing internal pictures of a subject. Three-dimensional diagnostic image data is acquired for respective “slices” of an area of the subject under investigation. These slices of data typically provide structural detail having a resolution of one (1) millimeter or better.
- the MR image pulse sequence includes magnetic field gradient waveforms, applied along three axes, and one or more RF pulses of energy.
- the set of gradient waveforms and RF pulses are repeated a number of times to collect sufficient data to reconstruct the slices of the image.
- the collected k-space data are typically reconstructed by performing an inverse Fourier transform (IFT).
- IFT inverse Fourier transform
- image reconstruction is not simple and artifacts, such as blurring due to off-resonance effects have to be corrected.
- 2D-FFTs have to be performed if the data set is large, which may cause impractical and unacceptable delays in image processing.
- sensing coils employed in MR image acquisition can have different and complex sensitivity profiles, which may make reconstruction from non-uniformly sampled k-space data impractical or at least difficult.
- the disclosed methods are directed at efficient MR image reconstruction for sensitivity encoding of non-rectilinearly-acquired MRI data, e.g., data acquired by spiral imaging.
- the systems and methods described herein are directed at a method of reconstructing a magnetic resonance image from non-rectilinearly-sampled k-space datasets obtained from associated magnetic resonance coils, the coils having associated sensitivity profiles.
- the method includes: for each coil, (a) distributing the associated sampled k-space dataset on a rectilinear k-space grid; (b) inverse Fourier transforming the distributed dataset; and (c) setting to zero a selected portion of the inverse-transformed dataset and retaining a remainder portion of the inverse-transformed dataset.
- the method further includes combining the remainder portions of the inverse-transformed datasets for the coils into a reconstructed magnetic resonance image.
- the method also includes: for each coil, modifying the reconstructed magnetic resonance image based at least partially on the sensitivity profile associated with the coil, to produce a modified dataset; Fourier transforming the modified dataset; at grid points associated with the selected zeroed portion, replacing the Fourier-transformed modified dataset with the distributed k-space dataset at corresponding points of the rectilinear k-space grid, to produce an updated dataset grid; for each coil, inverse Fourier transforming the updated dataset; and for each coil, until a difference between the inverse Fourier-transformed updated dataset and the inverse Fourier-transformed distributed dataset is sufficiently small, applying an iteration of steps b through h to the inverse Fourier-transformed updated dataset.
- the systems and methods described herein include a method of reconstructing a magnetic resonance image from non-rectilinearly-sampled k-space datasets obtained from associated magnetic resonance coils, the coils having associated sensitivity profiles.
- the method includes: for each coil, distributing the associated sampled k-space dataset on a rectilinear k-space grid; for each coil, convolving the distributed dataset with a sinc function; for each coil, and at least partially based on a characteristic of the sinc function, replacing a portion of the convolved dataset with a corresponding portion of the k-space dataset distributed on the rectilinear k-space grid, to produce an updated dataset; combining the updated datasets for the coils into a combined dataset; for each coil, modifying the combined dataset based at least partially on the sensitivity profile associated with the coil; for each coil, until a difference between the modified dataset and the distributed dataset is sufficiently small, applying an iteration of steps b through e to the modified dataset.
- FIG. 1 depicts a flow chart of a basic INNG algorithm.
- FIG. 2 depicts a flow chart of a facilitated INNG algorithm.
- FIG. 3 depicts an exemplary partition scheme of a Block INNG (BINNG) algorithm.
- FIG. 4 depicts an exemplary block diagram of Block Regional Off-Resonance Correction (BRORC).
- BORC Block Regional Off-Resonance Correction
- FIG. 5 depicts a flow chart of a POCSENSINNG algorithm.
- FIGS. 6( a )- 6 ( d ) depict carotid artery images acquired using four receiver channels and spiral trajectory data acquisition.
- FIGS. 7( a )- 7 ( d ) depict cardiac images acquired using four receiver channels and spiral trajectory data acquisition.
- FIGS. 8( a )- 8 ( b ) depict—respectively, before and after off-resonance correction—a selected cardiac region from an image reconstructed using the POCSENSINNG algorithm.
- the methods described herein are directed, inter alia, to efficient reconstruction of high-quality MR images.
- the methods described herein taking into account sensitivity profiles of coils used for MR data acquisition, can be applied to non-rectilinearly sampled data and spiral MRI sampling schemes.
- Non-rectilinear data acquisition methods have advantages over rectilinear data sampling schemes and hence are often performed in magnetic resonance imaging (MRI).
- MRI magnetic resonance imaging
- projection reconstruction i.e., radial trajectories
- spiral trajectories are insensitive to flow artifacts.
- Image reconstruction from non-rectilinearly sampled data is not simple, because 2D-Inverse Fourier Transform (IFT) cannot be directly performed on the acquired k-space data set.
- IFT 2D-Inverse Fourier Transform
- K-space gridding is commonly used as it is an efficient reconstruction method.
- Gridding is the procedure by which non-rectilinearly sampled k-space data are interpolated onto a rectilinear grid.
- the use of k-space gridding allows the reconstruction of images in general non-uniform sampling schemes, and thus gives flexibility to designing various types of k-space trajectories.
- BURS Block Uniform Resampling
- NNG Next Neighbor re-Gridding
- each acquired k-space datum is simply shifted to the closest grid point of a finer rectilinear grid than the original grid, in order to directly perform IFT on a non-uniformly sampled k-space in the NNG algorithm.
- the image quality of the NNG algorithm depends on the DCF used in step 1. In other words, non-negligible profile distortions of the reconstructed image are often observed if the DCF is not well optimized.
- the systems and methods described herein are directed at a new image reconstruction algorithm from non-rectilinearly sampled k-space data.
- the newly proposed algorithm is an extension of the NNG algorithm described above and will be referred to hereinafter as the ‘Iterative Next-Neighbor re-Gridding (INNG) algorithm’ as it includes an iterative approach.
- INN Next-Neighbor re-Gridding
- the algorithm requires a number of Fast Fourier Transforms (FFTs) of re-scaled matrices larger than the original-sized rectilinear grid matrix, no pre-calculated DCFs are required in the INNG algorithm, and the reconstructed image is of high quality.
- FFTs Fast Fourier Transforms
- BINNG Block INNG
- k-space is partitioned into several blocks and the INNG algorithm is applied to each block.
- data imperfections are non-negligible, e.g., low data SNR and/or a small scaling factor
- the background noise level in the reconstructed image is increased as the iteration progresses in the INNG/BINNG algorithms.
- the rate of the increase is usually not significant unless the data imperfections are substantial.
- the INNG/BINNG algorithms are a simple new approach to accurate image reconstruction and an alternative to the previously-proposed optimized gridding algorithms that does not require DCFs or SVD regularization parameter adjustments.
- the basic procedures of the INNG algorithm are presented as a flow chart.
- the originally-designed rectilinear grid size is N ⁇ N.
- the initial image of the INNG algorithm can be obtained by steps 2) and 3) in the Next-Neighbor re-Gridding (NNG) algorithm described above.
- NNG Next-Neighbor re-Gridding
- a 2D-IFT is performed on a large rescaled matrix after k-space data are distributed without density compensation.
- This procedure is equivalent to multiplication of the matrix (b) with a 2D-rect window function of amplitude 1 in the central N ⁇ N matrix and 0 elsewhere in the image. Therefore, if the matrix (c) is 2D-Fourier transformed, the obtained matrix (d) is the result of convolution of the matrix (a) with a 2D sinc function (which is 2D-FT of the 2D-rect function used in the previous process). After the matrix (d) is obtained, at the matrix coordinates where the original data exist in the rescaled matrix (a), the data are replaced by the original data values, as shown in the process (d) to (e) in FIG. 1 . Other matrix elements are left unchanged in this process.
- the Nyquist criterion is satisfied for the entire k-space region which spans from ⁇ k max to +k max along both k x and k y directions.
- at least one datum must exist in any s ⁇ s matrix region in the sN ⁇ sN rescaled matrix.
- the corresponding regions in the rescaled matrix are set to zeros. For example, in spiral trajectories, k-space regions outside of the circle with a radius
- the regions outside of the circle with a radius sN/2 are set to zeros in the sN ⁇ sN rescaled matrix, when the original data are inserted at each iteration.
- This procedure is also performed in the facilitated INNG algorithm and in the BINNG algorithm introduced in the following subsections.
- the INNG algorithm described above can be classified as a well-known optimization method ‘Projections Onto Convex Sets (POCS)’.
- POCS Projects Onto Convex Sets
- MRI the POCS method has been used in half-Fourier reconstruction, motion correction and parallel imaging reconstruction.
- each constraint can be formulated as a ‘convex set’, which is known in the art.
- two constraints are imposed on the data (or the image) at each iteration, that is, (i) the finite-support constraint and (ii) the data-consistency constraint.
- the constraints (i) and (ii) correspond to the process (b) to (c) and the process (d) to (e), respectively in FIG. 1 .
- I ( x out ) 0 ⁇ , [1] where I(x) is the image matrix of a large FOV (sN ⁇ sN) and x out represents all the matrix elements except the central N ⁇ N matrix.
- ⁇ 2 ⁇ I ( x )
- the constraint (i) is based on the signal sampling theory in which all the sampled signals must be expressed as the summation of rectilinearly located sinc functions. If all the data values in the large rescaled matrix can be expressed as the summation of the 2D sinc functions (each of which is the FT of the 2D-rect function with amplitudes 1 in the central N ⁇ N matrix and zero elsewhere), all the image matrix elements except the central N ⁇ N region must be zeros. The need for the constraint (ii) is to keep the original data values at the original data locations for each iteration.
- I m+1 ( x ) P 2 P 1 ⁇ I m ( x ) ⁇ . [3] where the subscript of I(x) denotes the iteration number.
- ⁇ x denotes summation of all the elements in the sN ⁇ sN image matrix.
- P 1 and P 2 are called non-expansive operators.
- the composite operator P 2 P 1 is also non-expansive, that is,
- the above iterative algorithm has a unique convergence point. However, if the errors contained in the data are non-negligible, a unique convergence point may not exist. Since both P 1 and P 2 are linear operators, P 2 P 1 is also a linear operator. Thus, the reconstructed image at the m-th iteration can be expressed as the summation of the image values that originate from the ideal signal components, i.e.
- Eq. [7] suggests that the image reconstructed using the basic INNG algorithm can be regarded as the summation of the image originated from the ideal signal components I m,ideal (x) and the image from the residual imperfections n m (x).
- the RMS energy of [I m,ideal (x) ⁇ I(x)] is continuously decreased toward zero, and that of n m (x) is increased.
- the increase rate of the RMS energy of n m (x) is reduced as the iteration progresses, since Eq. [6] holds for n m (x) as well.
- Eq. [7] leads to the ideal reconstructed image.
- the second term in Eq. [7] usually manifests itself as background noise in the reconstructed image, and the noise level is increased with iterations as will be seen in the ensuing section. However, the increased rate of the noise level is reduced as the iteration progresses.
- Eq.[6] also holds for n m (x). Therefore, it is expected that if the data SNR is within a practical range, and the scaling factor s is sufficiently large, the magnitude of I ideal,m (x) is still predominant over that of n m (x) after a certain number of iterations.
- ⁇ x N ⁇ N denotes summation of the central N ⁇ N image matrix elements.
- the iteration is stopped if d becomes lower than a predetermined value.
- the predetermined stopping criterion will be denoted by d s where the scaling factor is s in the following sections.
- the facilitated INNG algorithm modifies the basic INNG algorithm by employing consecutively increasing scaling factors.
- the image (a) is roughly close to the image reconstructed using the basic INNG algorithm with a larger scaling factor.
- a scaling factor of 8 is usually sufficient in practice to reduce data shift errors.
- an intermediate image reconstructed using one basic INNG algorithm is used as a starting image for the next basic INNG algorithm with a larger scaling factor.
- the final basic INNG algorithm must satisfy a rigorous stopping criterion, i.e., a small value of d in Eq.[8], in order to reconstruct a high-quality image, intermediate images do not have to satisfy a small d because they are merely ‘estimate images’ in the next basic INNG algorithm. Therefore, relaxed stopping criteria, i.e., relatively large d, can be used for all the basic INNG algorithms, with the exception of the last, in order to further improve the computational efficiency.
- the facilitated INNG algorithm substantially reduces the number of iterations from the basic INNG algorithm for the same target scaling factor s.
- intermediate images are used as starting images for the basic INNG algorithm with the next larger scaling factor, as shown in FIG. 2 .
- the number of iterations for the intermediate INNG algorithms could be reduced to further improve the computational efficiency.
- Stopping criteria used to obtain intermediate images
- Stopping criteria that are as relaxed as possible tend to avoid unnecessary computations, although it is typically difficult to determine the optimal stopping criteria in advance.
- FIG. 3 shows an exemplary partition scheme of the BINNG algorithm.
- the acquired k-space region is partitioned into several blocks, and the basic or facilitated INNG algorithm is applied to each block.
- the sampled k-space is partitioned into, for example, 3 ⁇ 3 blocks. All blocks do not need to be exactly the same size.
- the acquired k-space region is denoted as a square with its side length 2
- the basic INNG algorithm is applied to the shadowed block at the upper left corner in FIG. 3 .
- the scaling factor is s
- ) are distributed to an sN/2 ⁇ sN/2 matrix. Zero data values are assumed for the non-sampled k-space region within the bold square.
- the basic INNG algorithm is applied to the data within the bold square region using an sN/2 ⁇ sN/2 matrix as though the original target grid matrix size is N/2 ⁇ N/2.
- 2D-IFT is first performed on the sN/2 ⁇ sN/2 k-space data matrix (corresponding to (a) ⁇ (b) in FIG. 1 ), while zeros are set outside of the central N/2 ⁇ N/2 region (corresponding to (b) ⁇ (c) in FIG. 1 ).
- a 2D-FT is subsequently performed (corresponding to (c) ⁇ (d) in FIG. 1 ), and the original k-space data within the bold square region are inserted into the updated sN/2 ⁇ sN/2 data matrix (corresponding to (d) ⁇ (e) in FIG. 1 ).
- a 2D-IFT is then performed on the sN/2 ⁇ sN/2 data matrix (corresponding to (e) ⁇ (b) in FIG. 1 ). The above procedures are repeated until the difference between the updated matrix (b) and the matrix (b) at the previous iteration becomes sufficiently small. It is evident that an incomplete image appears in the central N/2 ⁇ N/2 region in the above iterations. However, both constraints (i) and (ii) of the INNG algorithm are effectively imposed on the sN/2 ⁇ sN/2 matrix at each iteration.
- the facilitated INNG algorithm can also be applied to the selected k-space data set by successively increasing the scaling factor.
- the extracted N/2 ⁇ N/2 matrix is transferred to the center of the next larger rescaled matrix of zeros after each basic INNG algorithm is performed.
- the obtained sN/2 ⁇ sN/2 data matrix may contain non-negligible errors in the regions close to the edges as the k-space data are abruptly truncated when they are selected. Therefore, only the part of the matrix that corresponds to the originally determined block (the shadowed region in FIG. 3 ) may be kept from the obtained sN/2 ⁇ sN/2 data matrix.
- an sN ⁇ sN k-space data matrix can be formed. It is expected that this data matrix satisfies both conditions (i) and (ii) for the entire region.
- a 2D-FFT is performed on the N ⁇ N data matrix obtained by s-fold decimation of the sN ⁇ sN data matrix.
- the sampled k-space region is partitioned into the exemplary 3 ⁇ 3 blocks, and the maximum size of the rescaled matrix is reduced to sN/2 ⁇ sN/2 from sN ⁇ sN required for the INNG algorithms.
- Other partition schemes and block sizes are also possible. For example, when the acquired k-space region is partitioned into 5 ⁇ 5 blocks, the maximum size of the rescaled matrix can be reduced to sN/4 ⁇ sN/4.
- partial Fourier reconstruction techniques can be employed to reduce scan time in spiral MR sampling schemes.
- this technique employs variable-density spiral (VDS) trajectories so that the Nyquist criterion is satisfied in the central region of the k-space, whereas the outer regions of k-space are undersampled.
- VDS variable-density spiral
- POCS projections onto convex sets
- PFSR partial Fourier spiral reconstruction
- the PFSR technique applies the projection onto convex sets (POCS) method (developed in rectilinear sampling schemes) to spiral sampling schemes.
- POCS convex sets
- the resealing matrix reconstruction algorithm (the analogous algorithm has been proposed as the INNG algorithm) has been modified.
- a first step of this algorithm includes creating an estimated image phase map ⁇ e from the low-resolution image reconstructed from the central k-space data.
- a next step is to perform iterative procedures to impose the two constraints on the acquired data set.
- the PFSR algorithm follows essentially the flow of the basic INNG algorithm described above with reference to FIG. 1 , except that in PFSR a phase constraint is imposed on the image (c) of FIG. 1 .
- the original target grid is an N ⁇ N matrix.
- the location of each datum in the large rescaled matrix is determined by multiplying the original k-space coordinate by s and then rounding the rescaled coordinate off to the nearest target rectilinear grid location. If more than one datum share the same matrix coordinate, the mean value is stored.
- IFT Inverse Fourier Transform
- a leading to image matrix (b).
- the intermediate reconstructed image appears in the central N ⁇ N matrix in (b).
- the region outside of the central N ⁇ N matrix is set to zeros, resulting in (c).
- An FT is performed on (c), leading to (d), which is an estimate of the phase-constrained raw data.
- the reconstructed image quality in the conventional rectilinear partial Fourier reconstruction with POCS has been shown to depend on the estimated phase, which is also the case with the PFSR algorithm discussed above.
- the variable-density spiral can sample the central region of k-space with little additional acquisition time as compared with a constant-density spiral.
- the estimated phase map can be efficiently obtained by using a VDS in the PFSR technique.
- Constraint (ii) is more difficult to apply when k-space data are sampled non-rectilinearly.
- the PFSR algorithm can overcome this difficulty, at least in part because it uses large rescaled (i.e., rectilinear K-space) matrices.
- constraints (i) and (ii) can be readily imposed on the data set at each of the iterations depicted in FIG. 1 with the phase constraint imposed in (c).
- the PFSR technique permits image reconstruction with reduced artifacts from undersampled spiral data sets, thereby enabling further reductions in scan time in spiral imaging.
- a Block Regional Off-Resonance Correction can be employed as a fast and effective deblurring method for spiral imaging.
- Spiral acquisition techniques have advantages over other k-space trajectories because of their short scan time and insensitivity to flow artifacts, but suffer from blurring artifacts due to off-resonance effects.
- a frequency-segmented off-resonance correction (FSORC) method is commonly used to combat off-resonance effects and reconstruct a deblurred image.
- FORC frequency-segmented off-resonance correction
- Deblurred image regions are selected from the reconstructed images under guidance of a frequency field map.
- the final reconstructed image with off-resonance correction is created by combining all deblurred regions selected from the appropriate demodulated image.
- the computational burden of FSORC is proportional to the number of demodulation frequencies used since the fast Fourier transform (FFT) is performed on each demodulated k-space data set.
- FFT fast Fourier transform
- MFI multi-frequency interpolation
- Image domain deconvolution methods approximate the spiral time evolution function as a quadratic function with respect to a k-space radius. This enables correction via one-dimensional deconvolution (along the x and y directions) in the image domain since separable demodulation functions along the x and y directions can be formed. However, image quality degradations beyond those associated with FSORC may result when the difference between the actual spiral time evolution function and the approximated quadratic function cannot be ignored.
- a novel fast off-resonance correction method (a.k.a., ‘Block regional off-resonance correction (BRORC)’) is presented.
- BRORC Block regional off-resonance correction
- FFTs are performed on matrices that are smaller than the full image matrix.
- the computational cost of BRORC relative to that of FSORC depends on the selection of the parameter values in these algorithms, the BRORC is usually computationally more efficient than FSORC.
- greater reduction of the computational costs can be expected in BRORC if only particular regions of the image need to be deblurred.
- a block diagram of BRORC having an original image matrix size of N ⁇ N (e.g., 256 ⁇ 256).
- the first step of the BRORC is to extract a small block region M ⁇ M.
- M is typically chosen to be a number expressed as a power of 2 (e.g., 16, 32), though this need not be the case.
- a 2D-FFT is performed on the M ⁇ M image matrix.
- the obtained M ⁇ M Fourier data is to be frequency demodulated.
- the demodulation function matrix for the M ⁇ M data must also be M ⁇ M in size. This matrix can be obtained by N/M-fold decimation of the original N ⁇ N demodulation function matrix.
- Regions near the four corners of the M ⁇ M demodulation function matrix should be handled carefully. Normally, after the acquired spiral k-space data are gridded onto an N ⁇ N grid, there are no data outside of the inscribed circle (radius N/2 in Cartesian step). These regions are usually set to zeros in the N ⁇ N data matrix before frequency demodulation is performed. However, in the M ⁇ M Fourier data matrix, all the M ⁇ M matrix elements usually have non-zero data values. If the corresponding M ⁇ M demodulation frequency matrix has zero values in the regions near the four corners, artifacts originating from the inaccurately demodulated high spatial frequency components may appear after demodulation. Therefore, when the M ⁇ M demodulation function matrix is formed, the regions outside the inscribed circle are filled with the maximum readout time values, thereby effectively performing N/M-fold decimation without introducing such artifacts.
- the demodulation frequency ('f indicated in FIG. 4 ) is determined from the central region of the M ⁇ M sub-image matrix in the frequency field map.
- the mean off-resonance frequency of the central rM ⁇ rM pixels (0 ⁇ r ⁇ 1, and r is typically 0.5.) in the M ⁇ M phase image matrix is used as the demodulation frequency 'f.
- the M ⁇ M k-space data is subsequently 2D-inverse Fourier transformed. Since the outer regions of the obtained M ⁇ M image matrix may exhibit artifacts, only the central rM ⁇ rM pixels of the M ⁇ M deblurred image matrix are kept for the final reconstructed image. This procedure is repeated until the entire scanned object is deblurred. However, as is evident from the BRORC block diagram, it is also possible to only deblur particular regions of the image. This is not possible with the conventional FSORC.
- a process flow of a POCSENSINNG algorithm is depicted, which extends the Projection-Onto-Convex-Sets (POCS) reconstruction method applied to sensitivity encoding with rectilinearly sampled data (SENSE), referred to as POCSENSE, to non-uniformly sampled data.
- POCS Projection-Onto-Convex-Sets
- SENSE rectilinearly sampled data
- POCSENSE rectilinearly sampled data
- k-space datasets are distributed to larger rescaled matrices, and the INNG algorithm is applied.
- the ‘POCENSINNG’ algorithm generally is a more efficient SENSE reconstruction algorithm from non-uniformly sampled k-space data.
- k-space data are obtained by n MRI sensing coils K 1 , K 2 , . . . , K n denoting n receiver channels, respectively.
- k-space coordinates in a sampled region over the range [ ⁇ k max , +k max ] are normalized to [ ⁇ N/2, +N/2].
- the sensing coils tend to have different sensitivity profiles, due, for example, to their design and/or spatial arrangement; it is desirable to take these sensitivities into account for the image analysis. If only a single coil is employed, then a single row in FIG. 5 would substantially represent the INNG process depicted in FIG. 1 , with the exception of step (c) in FIG. 5 .
- steps (c), (d) and (e) of FIG. 1 substantially correspond to steps (d), (e) and (f) in FIG. 5 . If the image matrix is scaled at each step by an increasing scaling factor s, then a single row in FIG. 5 would represent the Facilitated INNG process depicted in FIG. 2 .
- a location of each datum in the large rescaled matrix is determined by rounding off the original k-space coordinate after multiplying it by the scale factor s. If more than one datum share the same matrix coordinate, then, according to one practice, a mean value of the competing data is stored. In various embodiments, a variant of, or an alternative to, the mean is used; for example, a weighted mean or some other averaging measure may be employed. Inverse Fourier Transforms (IFFT) are performed on the matrices (a), leading to image matrices (b).
- IFFT Inverse Fourier Transforms
- Each of the g i 's in FIG. 5 denotes a corresponding image that appears in the central N ⁇ N region of a corresponding matrix; one such central region is depicted by 101 of FIG. 1 .
- one or more reconstructed image g i may be affected by aliasing artifacts due to undersampling.
- the g i 's are combined into g, as shown in (c) in FIG. 5 , using, for example and without limitation, the same method as that employed by POCSENSE:
- M represents an image mask.
- a method to create an image mask is described by, for example, Pruessman et al., “SENSE: Sensitivity Encoding for Fast MRI,” Mag. Res. Med., v. 42, pp. 952-962, 1999).
- Eq. [12] is one of the constraints of the POCSENSE and denoted as projection P 3 in Kholmovski et al., “POCSENSE: POCS-based reconstruction method for sensitivity encoded data”, Proceedings of the 10th Annual Meeting of ISMRM, , p. 194, Honolulu, 2002.
- the various central N ⁇ N images are combined into a single image g, as indicated in step (c) of FIG. 5 .
- the images are combined by using a modification of a method disclosed by the Kholmovski reference cited above.
- Kholmovski's method uses coil sensitivity profiles S i and alternating Projection Onto Convex Sets (POCS) formalisms to recover missing k-space data.
- POCS formulation of the problem permits natural incorporation of valid constraints (coil sensitivity profiles, acquired k-space data, and image support) in the reconstruction process as convex sets.
- Kholmovski's method does not require any type of computationally expensive matrix inversion operation and uses computationally efficient FFTs instead.
- Sensitivity profiles S i are assumed to have been previously determined for each coil.
- the sensitivity profile S i of i-th coil is multiplied by the single image g.
- the images S i g are then placed in the central N ⁇ N regions of sN ⁇ sN matrices, with the remaining matrix elements set to zero (d).
- FFTs are then performed on the matrices (d), which transform the matrices (d) back into k-space, leading to new estimates G i of the rescaled k-space data, step (e).
- the original k-space data (prior to being zeroed out) from the rescaled matrices K i (as shown in (a)) are inserted into the rescaled matrices K i at their original locations, as shown in (f) and described above with reference to step (e) of FIG. 1 ; data in other locations are unaltered.
- IFFTs are then performed on the matrices (f).
- the updated spatial-domain images g i appear in the central N ⁇ N regions on the matrices (b).
- the procedures (b) ⁇ (c) ⁇ (d) ⁇ (e) ⁇ (f) ⁇ (b) (surrounded by dashed lines in FIG.
- a large scaling factor s would be desirable to reduce the data shift errors in the large rescaled matrices.
- a final scaling factor of 4 is usually sufficient in practice for in-vivo MR image reconstruction.
- the POCSENSINNG algorithm was applied to in-vivo MR data that were acquired with spiral trajectories. Data acquisitions were performed using a 1.5-Tesla Siemens Sonata scanner (Siemens Medical Solutions, Er Weg, Germany). Carotid arteries were scanned from an asymptomatic volunteer using four-element surface coils. Cardiac images were also acquired from another asymptomatic volunteer using four-element phased array torso/body surface coils. All procedures were performed under an institutional review board-approved protocol for volunteer scanning.
- TI was set to 700 ms.
- 20 spiral interleaves were used with a field of view (FOV) 170 ⁇ 170 mm.
- Slice thickness 10 mm, spiral readout time 16.0 ms, and TE/TR 6.6/2000.0 ms.
- 1-2-1 binomial pulses were used for spatial-spectral excitation.
- Two spiral interleaves were successively acquired in one TR.
- the total flip angles for on-resonance spins were 45° (1st) and 90° (2nd).
- T2 preparation pulses were used (7).
- the length of T2 prep pulses was set to 30 ms.
- ECG gating was used during the acquisitions.
- 1-2-1 binomial pulses were used for spatial-spectral excitation.
- Two spiral interleaves were successively acquired for one cardiac cycle.
- the total flip angles for on-resonance spins were 45° (1st) and 90° (2nd).
- FIG. 6 shows the carotid artery images.
- the images were reconstructed from the spiral data of 10 interleaves out of 20 acquired interleaves ((a): the sum-of squares method; (b): the POCSENSINNG algorithm after 1st iteration; (c): the POCSENSINNG algorithm after 10th iteration).
- the image was reconstructed using the sum-of-squares method from the spiral data of all 20 acquired interleaves.
- spiral aliasing artifacts are observed for the entire image region.
- low spatial frequency components predominantly appear in the reconstructed image after 1st iteration in the POCSENSINNG algorithm.
- FIG. 7 shows the cardiac images.
- the amounts of spiral interleaves used for image reconstruction and the reconstruction methods were the same as (a)-(d) in FIG. 6 , respectively.
- the image is affected by substantial level of spiral aliasing artifacts.
- low spatial frequency components predominantly appear in the POCSENSINNG image after 1st iteration.
- the image after 10th iteration (c) exhibits sufficient amounts of high frequency components.
- the noise levels of image (c) appear higher than that of full data set image (d) because of reduced data acquisition, the aliasing artifacts observed in (a) are effectively reduced in (c).
- FIG. 8( a ) shows the selected 128 ⁇ 128 region of cardiac image FIG. 7( c ).
- FIG. 8( b ) is the same image region after off-resonance correction. Spiral off-resonance blurring artifacts are effectively reduced after off-resonance correction. For example, as seen in the regions indicated by arrows, the edge definitions of aorta and main pulmonary artery are improved after off-resonance correction.
- the data-consistency constraint required for POCSENSE is difficult to apply when k-space data are acquired non-uniformly.
- this constraint can be readily imposed on the data sets in the POCSENSINNG algorithm as it takes advantage of large rescaled matrices.
- the previously proposed Conjugate Gradient (CG) iteration method performs gridding operations for both forward (k-space to image) and reverse (image to k-space) directions as well as sampling density compensation at each iteration (2).
- the POCSENSINNG algorithm obviates the need for these complicated gridding procedures; it simply inserts the acquired k-space data into corresponding locations in large rescaled matrices at each iteration.
- the INNG algorithms do not require density compensation and leads to accurate image reconstruction with sufficiently large scaling factor (5).
- the POCSENSINNG algorithm takes advantages of this characteristic of the INNG algorithms.
- FFT ((d) ⁇ (e)) and IFFT ((f) ⁇ (b)) are the most computationally intensive parts in the POCSENSINNG algorithm.
- the computational costs of other operations e.g., combine images g i to g ((b) ⁇ (c)), multiply g by S i ((c) ⁇ (d)), insert K i to G i ((e) ⁇ (f))
- the reconstruction time mainly depends on the size of rescaled matrices and the number of FFT to be performed in the POCSENSINNG algorithm. For example, in the image reconstruction of FIGS. 6 and 7 (512 ⁇ 512 rescaled matrices and 4 receiver channels), one iteration took about 3.3 s in a workstation with an Intel Pentium IV processor operating at 1.70 GHz and 512 MB RAM.
- the facilitated INNG algorithm can be utilized in the POCSENSINNG algorithm to reduce the number of iterations. In other words, s is consecutively increased during the iterations.
- the computation time is usually significantly increased when FFTs are performed on the rescaled matrices with a large s. This can be understood from the fact that the total number of complex multiplications required for sN ⁇ sN 2D-FFT is 2 (sN) 2 log 2 (sN).
- sN the total number of complex multiplications required for sN ⁇ sN 2D-FFT is 2 (sN) 2 log 2 (sN).
- sN 2 log 2
- the background is forced to be zeros in the combined image g at each iteration in the POCSENSINNG algorithm. Therefore, the image signal-to-noise ratio (SNR) is difficult to evaluate in the image reconstructed using the POCSENSINNG algorithm because the background noise level is usually significantly low.
- the data shift errors in the large rescaled matrices usually result in the reduction of image SNR in the INNG algorithms ( 5 ). In this example, the noise level in cardiac region in FIG. 7( c ) appear higher than that in the same region of the full data set image FIG. 7( d ).
- off-resonance correction needs to be performed only once after the image is reconstructed.
- off-resonance correction may be incorporated in the gridding operations of CG procedures ( 2 ).
- spiral off-resonance correction can be performed after the iterations are complete in the POCSENSINNG algorithm also speeds up the reconstruction when off-resonance artifacts need to be corrected in spiral imaging.
- the proposed POCSENSINNG algorithm is an extension of POCSENSE to non-uniform sampling schemes using INNG algorithms. It takes advantage of large rescaled matrices and thus all the constraints necessary for POCSENSE can be readily imposed at each iteration. It also avoids the need for complicated gridding procedures.
- the POCSENSINNG algorithm is a useful practical reconstruction algorithm for sensitivity encoded data with arbitrary k-space trajectories.
Abstract
Description
Ω1 ={I(x)|I(x out)=0}, [1]
where I(x) is the image matrix of a large FOV (sN×sN) and xout represents all the matrix elements except the central N×N matrix.
Ω2 ={I(x)|I(x)=F{D(n)},D(n orig)=D orig(n orig)}, [2]
where F is the Fourier Transform operator, D(n) is the Fourier data matrix (sN×sN) of I(x), norig represents all the elements in the larger scaled matrix where the original data exist, and Dorig are the original data values at these coordinates.
I m+1(x)=P 2 P 1 {I m(x)}. [3]
where the subscript of I(x) denotes the iteration number.
where
denotes summation of all the elements in the sN×sN image matrix. P1 and P2 are called non-expansive operators. The composite operator P2P1 is also non-expansive, that is,
Note that Eqs. [4,5,6] hold whether or not the data are ideal. The algorithms with non-expansive operators have certain convergence properties. If the data distributed in the larger rescaled matrix are ideal, then the above iterative algorithm has a unique convergence point. However, if the errors contained in the data are non-negligible, a unique convergence point may not exist. Since both P1 and P2 are linear operators, P2P1 is also a linear operator. Thus, the reconstructed image at the m-th iteration can be expressed as the summation of the image values that originate from the ideal signal components, i.e. the signal components which satisfy the condition (i) Iideal,m(x) and the image values that originate from the residual imperfect signal components nm(x):
I m(x)=I ideal,m(x)+n m(x)(=P 2 P 1 {I ideal,m−1(x)}+P 2 P 1 {n m−1(x)}) [7]
monotonically decreases with iteration number m. In the present embodiment, the sum of the squared difference [Im(x)−Im+1(x)]2 is calculated within the central N×N image matrix instead of the entire sN×sN image matrix to facilitate the computation. Hence, the following quantity d is measured to determine where to stop the iteration:
where
denotes summation of the central N×N image matrix elements. The iteration is stopped if d becomes lower than a predetermined value. The predetermined stopping criterion will be denoted by ds where the scaling factor is s in the following sections.
I new =|I old|*exp(iΦ e), [9]
where Iold and Inew represent the image values at each pixel in the central N×N region of (b) before and after the phase constraint, respectively. The region outside of the central N×N matrix is set to zeros, resulting in (c). An FT is performed on (c), leading to (d), which is an estimate of the phase-constrained raw data. Then, a data-consistency constraint is imposed on this data matrix, i.e., the data where the original data exist are replaced by the original data values, as shown in (e). An IFT is performed on (e). The updated reconstructed image again appears in the central N×N matrix (b). The procedures (b)→(c)→(d)→(e)→(b) (surrounded by dashed lines in
where σi is the noise standard deviation in i-th channel, Si is the sensitivity profile of i-th coil and Si* is its complex conjugate. After g is created using Eqs. [10-11], the image values for the region of support are maintained and those for other regions are set to zeros, i.e.,
where M represents an image mask. A method to create an image mask is described by, for example, Pruessman et al., “SENSE: Sensitivity Encoding for Fast MRI,” Mag. Res. Med., v. 42, pp. 952-962, 1999). Eq. [12] is one of the constraints of the POCSENSE and denoted as projection P3 in Kholmovski et al., “POCSENSE: POCS-based reconstruction method for sensitivity encoded data”, Proceedings of the 10th Annual Meeting of ISMRM, , p. 194, Honolulu, 2002.
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